Codeforces 1139D（推式子+dp）

题目传送
推公式博客传送

推完式子就是去朴素地求就行了Orz

const int maxn = 1e5 + 5;
const int mod = 1e9 + 7;
int m, mu[maxn], vis[maxn], primes[maxn], tot;
ll dp[maxn];
vector<int> factor[maxn];

ll ksm(ll a, ll b) {
	ll ret = 1;
	for (; b; b >>= 1) {
		if (b & 1)	ret = ret * a % mod;
		a = a * a % mod;
	}
	return ret;
}

void pre(int n) {
	rep(i, 1, n) {
		for (int k = 1; k * i <= n; k++)
			factor[k * i].push_back(i);
	}

	mu[1] = 1;
	rep(i, 2, n) {
		if (!vis[i]) {
			primes[tot++] = i;
			mu[i] = mod - 1;
		}
		for (int j = 0; j < tot && primes[j] * i <= n; j++) {
			vis[primes[j] * i] = 1;
			if (i % primes[j] == 0)	break;
			mu[primes[j] * i] = mod - mu[i];
		}
	}
}

ll calc(int x, int d) {
	ll ret = 0;
	for (int i : factor[x / d]) {
		ret = (ret + (ll)mu[i] * (m / d / i) % mod) % mod;
	}
	return ret;
}

void DP() {
	dp[1] = 0;
	rep(i, 2, m) {
		ll k = m - m / i;
		ll tmp = m;
		for (auto j : factor[i]) {
			if (j == i)	continue;
			tmp = (tmp + dp[j] * calc(i, j) % mod) % mod;
		}
		dp[i] = tmp * ksm(k, mod - 2) % mod;
	}
}

ll ANS(int m) {
	ll ret = 0;
	rep(i, 1, m) {
		ret = (ret + dp[i] + 1) % mod;
	}
	return ksm(m, mod - 2) * ret % mod;
}

int main() {
	read(m);
	pre(m);
	DP();
	writeln(ANS(m));
	return 0;
}